Question: Find the least common multiple $(\text{LCM})$ of $13k^2+26k+13$ and $5k^5+25k^4+20k^3$. You can give your answer in its factored form.
Answer: The least common multiple $(\text{LCM})$ of two polynomial expressions is the polynomial with the least number of factors that is divisible by both polynomials. [How does this relate to the least common multiple of integers?] We can find the $\text{LCM}$ by factoring the two polynomials as much as possible and then comparing the factors: $13k^2+26k+13$ can be factored as ${(13)}{(k+1)}{(k+1)}$ by factoring out a $13$ and using the perfect square pattern. $5k^5+25k^4+20k^3$ can be factored as ${(5)(k^3)}{(k+1)}{(k+4)}$ by factoring out a $5k^3$ and using the sum-product pattern. We can see that: Both polynomials share the factors ${(k+1)}$ Only the first polynomial has the factors ${(13)}{(k+1)}$ Only the second polynomial has the factors ${(5)(k^3)(k+4)}$ Therefore, the least common multiple is the product of all the above factors: [Why?] $\begin{aligned}&\phantom{=}{(k+1)}{(13)}{(k+1)}{(5)(k^3)(k+4)}\\\\ &=65k^3(k+1)^2(k+4)\end{aligned}$ In conclusion, the least common multiple of the two polynomials is $65(k^3)(k+1)^2(k+4)$.